# Some estimates for $\theta$-type Calder\'on-Zygmund operators and linear commutators on certain weighted amalgam spaces

## TL;DR

This paper introduces new weighted amalgam spaces and establishes strong and weak type estimates for $ heta$-type Calderón-Zygmund operators and their linear commutators within these spaces, including endpoint and two-weight inequalities.

## Contribution

The paper develops novel weighted amalgam spaces and provides new boundedness results for Calderón-Zygmund operators and their commutators on these spaces.

## Key findings

- Established strong and weak type estimates for $T_ heta$ and $[b,T_ heta]$
- Proved endpoint estimates for linear commutators
- Analyzed two-weight weak type inequalities in weighted amalgam spaces

## Abstract

In this paper, we first introduce some new kinds of weighted amalgam spaces. Then we discuss the strong type and weak type estimates for a class of Calder\'on--Zygmund type operators $T_\theta$ in these new weighted spaces. Furthermore, the strong type estimate and endpoint estimate of linear commutators $[b,T_{\theta}]$ formed by $b$ and $T_{\theta}$ are established. Also we study related problems about two-weight, weak type inequalities for $T_{\theta}$ and $[b,T_{\theta}]$ in the weighted amalgam spaces and give some results.